AI 中文总结
研究零阶牛顿动力学，开发从黑箱函数值估计梯度和海森矩阵的动力学框架，用高斯 - 斯坦因校正偏差，线性化逆海森矩阵揭示噪声通道，通过小质量动力学提升联系模型，数值实验证实相关定律和行为。
AI 中文摘要
零阶牛顿型方法在梯度和海森矩阵不可用时很有用，但与一阶无梯度方法行为不同。我们为从黑箱函数值估计梯度和海森矩阵的算法开发了一个动力学框架。朴素的随机方向海森矩阵估计器即使在二次函数上也有偏差，需要高斯 - 斯坦因校正来估计高斯平滑目标的海森矩阵。线性化逆海森矩阵暴露出两个噪声通道。通过小质量动力学提升将有限步牛顿更新与欠阻尼相空间模型联系起来，过阻尼空间极限产生一个李雅普诺夫界，揭示了步长、批量大小、平滑半径和正则化之间的曲率 - 方差权衡。数值实验证实了估计器恒等式、梯度和海森矩阵方差定律、维度缩放、逆扰动精度以及在查询预算和正则化消融下的优化行为。
英文摘要
Zeroth-order Newton-type methods are useful when gradients and Hessians are unavailable, but they behave quite differently from first-order gradient-free methods. We develop a kinetic framework for algorithms that estimate both gradient and Hessian from black-box function values. The naive random-direction Hessian estimator turns out to be biased even on quadratics; a Gaussian--Stein correction is needed to estimate the Hessian of the Gaussian-smoothed objective. Linearizing the inverse Hessian exposes two noise channels: gradient noise preconditioned by the inverse Hessian, and Hessian noise transmitted through an inverse-Hessian sandwich. Under a noisy oracle the second channel carries the second-difference factor $μ_H^{-4}$. A small-mass kinetic lift links the finite-step Newton update to an underdamped phase-space model; the overdamped spatial limit yields a Lyapunov bound that exposes the curvature--variance trade-off between step size, batch sizes, smoothing radii, and regularization. Numerical experiments confirm estimator identities, the gradient and Hessian variance laws, dimension scaling, inverse-perturbation accuracy, and optimization behavior under query-budget and regularization ablations.
CommentsWithdrawn due to serious concerns regarding the authenticity and accuracy of the listed authorship. The identity of one or more listed authors cannot presently be verified, and the author list may not represent distinct contributors. The manuscript is withdrawn pending institutional review
